# Greatest lower bound

## Main Question or Discussion Point

prove: "A nonempty set of real numbers bounded from below has a greatest lower bound."

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cristo
Staff Emeritus
This looks like homework. What thoughts do you have on the question? What is the definition of the greatest lower bound of a set?

A lower bound of a non-empty subset A of R is an element d in R with d <= a for all a A.
An element m in R is a greatest lower bound or infimum of A if
m is a lower bound of A and if d is an upper bound of A then m >= d.

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cristo
Staff Emeritus
Ok, so here A is bounded below, so this tells you that there exists a lower bound to A. It may now be helpful to consider the set
-A:={-x:x∈A}, and use the completeness axiom to find the least upper bound on -A. The relationship between A and -A should help you find the greatest lower bound of A.

this is not number theory

this is not number theory, but it is instead mathematical analysis.

it is a very basic result that has been used to construct the real numbers and i think you will find it in any standard intro to analysis textbook (i personally recommend principles of mathematical analysis by Walter Rudin).

hope it helps

HallsofIvy
Homework Helper
Mariouma and Cristo are assuming that you are allowed to use the fact that "if a set of real numbers has an upper bound then it has a least upper bound"- what Cristo called the "completeness axiom". Is that true?

adityab88 is, I think, assuming that you have to prove that completenss axiom from the definition of the real numbers. It very easy to do that from the Dedekind cut definition of the real numbers.

dedekind cuts/ cauchy sequences

Mariouma and Cristo are assuming that you are allowed to use the fact that "if a set of real numbers has an upper bound then it has a least upper bound"- what Cristo called the "completeness axiom". Is that true?

adityab88 is, I think, assuming that you have to prove that completenss axiom from the definition of the real numbers. It very easy to do that from the Dedekind cut definition of the real numbers.
precisely. dedekind cuts or even cauchy sequences can be used prove such a statement.